Assume that \(\left( \Phi _{(n-1)f}^{\ast }(p),\left|X_{x}\right|\right) =1\) . Then \(X\)  is isomorphic either to \(\Omega _{n}(q)\)  with \(nq\)  odd, or to \(PSU_{n}(q^{1/2})\)  with \(n\)  even. Suppose that
the latter occurs. Then \(X_{B}^{^}\)  fixes a non-degenerate 1-dimensional subspace of \(V_{n}(q)\)  by Proposition REF (2) and by [1]}, Propositions
4.1.4 and 4.1.20, since its order is divisible by \(\Phi _{(n-1)f}^{\ast }(p)^{1/2}\) . Hence \(X_{B}^{^}\leqslant GU_{n-1}(q^{1/2})\) . Then \(X_{B}^{^}\cap SU_{n-1}(q^{1/2})\)  is a subgroup of \(SL_{n-1}(q)\) 
such that \(\Phi _{n-1f}^{\ast }(p)\)  divides the order of \(X_{B}^{^}\cap SU_{n-1}(q^{1/2})\)  and the index of this one in \(SL_{n-1}(q)\) . Also \(p\) 
divides the order of \(X_{B}^{^}\cap SU_{n-1}(q^{1/2})\) . Then we
may apply Proposition REF , with \(X_{B}^{^}\cap SU_{n-1}(q^{1/2})\)  in the role of \(Y\)  and \(SL_{n-1}(q)\)  in the role of \(SL_{n}(q)\) , and we obtain that \(X_{B}^{^}\cap SU_{n-1}(q^{1/2})\)  is contained in \(GL_{1}(q^{n-1})\cdot Z_{n-1}\) , with \(p \mid n-1\) , since \(q\)  is a square. Then \(\left|X_{B}^{^}\right|_{p}\leqslant (n-1)_{p}\leqslant n-1\) , as \(\left|X_{B}^{^}\cap SU_{n-1}(q^{1/2})\right|_{p}=\left|X_{B}^{^}\right|_{p}\) . On the other hand, \(\left|X_{B}\right|_{p}\geqslant q^{\frac{\left( n+2\right) \left( n-3\right) }{12}}\)  by Corollary REF (iii). Therefore \(q^{\frac{\left( n+2\right) \left( n-3\right) }{12}}\leqslant n-1\) 
and hence \(n=4\)  and \(q=4\)  or 9, but in none of these cases \(\Phi _{2(n-1)}^{\ast }(p)\)  is a square.
Assume that \(X\cong \)  \(\Omega _{n}(q)\)  with \(nq\)  odd. Then \(X_{B}^{^}\)  preserves a non-degenerate 1-dimensional subspace of \(V_{n}(q)\)  by
Proposition REF  and by [1]}, Propositions 4.1.6 and 4.1.14, since \(\Phi _{(n-1)f}^{\ast }(p)^{1/2}\)  divides \(\left( \left|X_{B}^{^}\right|,\left[ SL_{n}(q):X_{B}^{^}\right] \right)\)  and since \(p\)  divides \(\left|X_{B}^{^}\right|\)  by Corollary REF . Thus \(X_{B}^{^}\leqslant \Omega _{n-1}^{-}(q).Z_{2}<SL_{n-1}(q)\)  and hence \(X_{B}^{^}\)  is a subgroup of \(SL_{n-1}(q)\)  such that \(\Phi _{(n-1)f}^{\ast }(p)^{1/2}\)  divides \(\left( \left|X_{B}^{^}\right|,\left[ SL_{n-1}(q):X_{B}^{^}\right] \right) \) . Also \(p\mid \left|X_{B}^{^}\right|\) .
Then we may apply Proposition REF , with \(X_{B}^{^}\)  in the
role of \(Y\)  and \(SL_{n-1}(q)\)  in that of \(SL_{n}(q)\) , thus obtaining either \(q=7\) , \(n-1=4\)  and \(Z_{2}.A_{7} \unlhd X_{B}^{^} \leqslant Z_{2}.A_{7}\) , or \(X_{B}^{^}\leqslant GL_{1}(q^{n-1})\cdot Z_{n-1}\)  and \(p\mid n-1\) . The former case cannot occur, since \(X \cong \Omega _{5}(7) \cong \mathrm {PSp}_{4}(7)\)  contains \(A_{7}\)  but not its covering group \(Z_{2}.A_{7}\) . Thus \(X_{B}^{^}\leqslant GL_{1}(q^{n-1})\cdot Z_{n-1}\)  and \(p\mid n-1\) , and hence \(\left|X_{B}\right|_{p}\leqslant (n-1)_{p}\leqslant n-1\) . On the other hand, \(q^{\frac{(n-1)^{2}-12}{12}}\leqslant \left|X_{B}\right|_{p}\)  by Corollary REF (3). Then \(q^{\frac{(n-1)^{2}-12}{12}}\leqslant n-1\)  and hence \(n=5\)  and \(q<4^{3}\) , since \(n\geqslant 5\)  by Remark, REF . Then \(q=7\) , since \(\Phi _{4f}^{\ast }(p)\)  must be a square, but 7 does not divide \(n-1=4\) . So, this case cannot occur and the proof is completed.
